Eigenvalue distribution of large sample covariance matrices of linear processes

• Autorzy: Pfaffel O. , Schlemm E.
• Języki publikacji: EN

Abstrakty

EN

We derive the distribution of the eigenvalues of a large sample covariance matrix when the data is dependent in time. More precisely, the dependence for each variable i = 1, . . . , p is modelled as a linear process (X_i;t)_t=1...,n = (Σ^∞_j=0 c_jZ_i;t−j )_t=1,...,n, where {Z_i;t} are assumed to be independent random variables with finite fourth moments. If the sample size n and the number of variables p = p_n both converge to infinity such that y = limn→∞ n/p_n > 0, then the empirical spectral distribution of p⁻¹XX^T converges to a non-random distribution which only depends on y and the spectral density of (X_1;t)_t∈Z. In particular, our results apply to (fractionally integrated) ARMA processes, which will be illustrated by some examples.

Słowa kluczowe

EN

eigenvalue distribution; fractionally integrated; ARMA process; limiting spectral distribution; linear process; random matrix theory; sample covariance matrix

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2011
• Tom: Vol. 31, Fasc. 2
• Strony: 313--329
• Opis fizyczny: Bibliogr. 20 poz., wykr.

Twórcy

autor

Pfaffel O.

• Technische Universität München, Boltzmannstraße 3, 85748 Garching, Germany

autor

Schlemm E.

• Technische Universität München, Boltzmannstraße 3, 85748 Garching, Germany

Bibliografia

• [1] G. W. Anderson, A. Guionnet and O. Zeitouni, An Introduction to Random Matrices, Cambridge Stud. Adv. Math., Vol. 118 (2010).
• [2] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, third edition, Wiley Ser. Probab. Statist., 2003.
• [3] Z. Bai and J. W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, second edition, Springer Ser. Statist., 2010.
• [4] Z. D. Bai , D. Jiang, J .-F. Yao and S. Zheng, Corrections to LRT on large-dimensional covariance matrix by RMT, Ann. Statist. 37 (6B) (2009), pp. 3822-3840.
• [5] Z. D. Bai and W. Zhou, Large sample covariance matrices without independence structures in columns, Statist. Sinica 18 (2) (2008), pp. 425-442.
• [6] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, second edition, Springer Ser. Statist., 1991.
• [7] Z. Burda, A. Jarosz, M. A. Nowak and M. Snarska, A random matrix approach to VARMA processes, New J. Phys. 12 (2010), 075036.
• [8] S. Gerschgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Bull. Acad. Sci. URSS. Cl. sci. math. nat. 6 (1931), pp. 749-754.
• [9] C. W. J. Granger and R. Joyeux, An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal. 1 (1) (1980), pp. 15-29.
• [10] R. M. Gray, Toeplitz and Circulant Matrices: A Review, Now Publishers, Boston 2006.
• [11] U. Grenander and G. Szegő, Toeplitz Forms and Their Applications, second edition, Chelsea Publishing, New York 1984.
• [12] J. R. M. Hosking, Fractional differencing, Biometrika 68 (1) (1981), pp. 165-176.
• [13] Iain M. Johnstone, On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist. 29 (2) (2001), pp. 295-327.
• [14] V. A. Marchenko and L. A. Pastur, Distribution of eigenvalues in certain sets of random matrices, Mat. Sb. (N.S.) 72 (114) (4) (1967), pp. 507-536.
• [15] M. L. Mehta, Random Matrices, third edition, Pure Appl. Math., Elsevier/Academic Press, Amsterdam 2004.
• [16] G. Pan, Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix, J. Multivariate Anal. 101 (6) (2010), pp. 1330-1338.
• [17] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr and H. E. Stanley, Random matrix approach to cross correlations in financial data, Phys. Rev. E 65 (6) (2002), 66126.
• [18] M. Potters, J.-P. Bouchaud and L. Laloux, Financial applications of random matrix theory: old laces and new pieces, Acta Phys. Polon. B 36 (9) (2005), pp. 2767-2784.
• [19] A. M. Tulino and S. Verdu, Random Matrix Theory and Wireless Communications, Now Publishers, Boston 2004.
• [20] J. Wishart, The generalised product moment distribution in samples from a normal multivariate population, Biometrika 20A (1/2) (1928), pp. 32-52.